Thermo-physical properties of water and nanoparticles [39].

## 1. Introduction

The natural convection flow over a surface embedded in saturated porous media is encountered in many engineering problems such as the design of pebble-bed nuclear reactors, ceramic processing, crude oil drilling, geothermal energy conversion, use of fibrous material in the thermal insulation of buildings, catalytic reactors and compact heat exchangers, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, storage of nuclear wastes, etc.

The derivation of the empirical equations which govern the flow and heat transfer in a porous medium has been discussed in [1-5]. The natural convection on vertical surfaces in porous media has been studied used Darcy’s law by a number of authors [6–20]. Boundary layer analysis of natural convection over a cone has been investigated by Yih [21-24]. Murthy and Singh [25] obtained the similarity solution for non-Darcy mixed convection about an isothermal vertical cone with fixed apex half angle, pointing downwards in a fluid saturated porous medium with uniform free stream velocity, but a semi-similar solution of an unsteady mixed convection flow over a rotating cone in a rotating viscous fluid has been obtained Roy and Anilkumar [26]. The laminar steady nonsimilar natural convection flow of gases over an isothermal vertical cone has been investigated by Takhar et al. [27]. The development of unsteady mixed convection flow of an incompressible laminar viscous fluid over a vertical cone has been investigated by Singh and Roy [28] when the fluid in the external stream is set into motion impulsively, and at the same time the surface temperature is suddenly changed from its ambient temperature. An analysis has been carried out by Kumari and Nath [29] to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection. Cheng [30-34] focused on the problem of natural convection from a vertical cone in a porous medium with mixed thermal boundary conditions, Soret and Dufour effects and with variable viscosity.

The conventional heat transfer fluids including oil, water and ethylene glycol etc. are poor heat transfer fluids, since the thermal conductivity of these fluids play an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. An innovative technique for improving heat transfer by using ultra fine solid particles in the fluids has been used extensively during the last several years. Choi [35] introduced the term nanofluid refers to these kinds of fluids by suspending nanoparticles in the base fluid. Khanafer et al. [36] investigated the heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The convective boundary-layer flow over vertical plate, stretching sheet and moving surface studied by numerous studies and in the review papers Buongiorno [37], Daungthongsuk and Wongwises [38], Oztop [39], Nield and Kuznetsov [40,41], Ahmad and Pop [42], Khan and Pop [43], Kuznetsov and Nield [44,45] and Bachok et al. [46].

From literature survey the base aim of this work is to study the free convection boundary-layer flow past a vertical cone embedded in a porous medium filled with a nanofluid, the basic fluid being a non-Newtonian fluid by using similarity transformations. The reduced coupled ordinary differential equations are solved numerically. The effects of the parameters governing the problem are studied and discussed.

## 2. Mathematical formulation of the problem

Consider the problem of natural convection about a downward-pointing vertical cone of half angle

The nanofluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermo physical properties of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [39]). Assuming that the thermal boundary layer is sufficiently thin compared with the local radius, the equations governing the problem of Darcy flow through a homogeneous porous medium saturated with power-law nanofluid near the vertical cone can be written in two-dimensional Cartesian coordinates

Where

Here

The associated boundary conditions of Eqs. (1)-(3) can be written as:

where

Pure water | 997.1 | 4179 | 0.613 | 21 |

Copper (Cu) | 8933 | 385 | 401 | 1.67 |

Silver (Ag) | 10500 | 235 | 429 | 1.89 |

Alumina (Al_{2}O_{3}) | 3970 | 765 | 40 | 0.85 |

Titanium Oxide (TiO_{2}) | 4250 | 686.2 | 8.9538 | 0.9 |

By introducing the following non-dimensional variables:

The continuity equation is automatically satisfied by defining a stream function

where;

Integration the momentum Eq. (2) we have:

Substituting variables (6) into Eqs. (1)–(5) with Eq. (9), we obtain the following system of ordinary differential equations:

along with the boundary conditions:

where primes denote differentiation with respect to

where

## 3. Results and discussion

In this study we have presented similarity reductions for the effect of a nanoparticle volume fraction on the free convection flow of nanofluids over a vertical cone via similarity transformations. The numerical solutions of the resulted similarity reductions are obtained for the original variables which are shown in Eqs. (10) and (11) along with the boundary conditions (12) by using the implicit finite-difference method. The physical quantity of interest here is the Nusselt number _{2}O_{3}) and Titanium oxide (TiO_{2}).

In order to verify the accuracy of the present method, we have compared our results with those of Yih [22] for the rate of heat transfer

Vertical plate | Vertical cone | |||

Yih [22] | Present method | Yih [22] | Present method | |

0.5 | 0.3766 | 0.3768 | 0.6522 | 0.6524 |

0.8 | 0.4237 | 0.4238 | 0.7339 | 0.7340 |

1.0 | 0.4437 | 0.4437 | 0.7686 | 0.7686 |

1.5 | 0.4753 | 0.4752 | 0.8233 | 0.8233 |

2.0 | 0.4938 | 0.4938 | 0.8552 | 0.8552 |

0.05 | 0.7423 | 0.7704 | 0.6604 | 0.6725 |

0.1 | 0.6931 | 0.7330 | 0.5642 | 0.5852 |

0.15 | 0.6301 | 0.6732 | 0.4780 | 0.5057 |

0.2 | 0.5591 | 0.6002 | 0.4006 | 0.4331 |

0.3 | 0.4052 | 0.4357 | 0.2673 | 0.3062 |

Figs. 5 and 6 are presented to show the effect of the volume fraction of nanoparticles Cu and Ag respectively, on temperature distribution. These figures illustrate the streamline for different values of

Fig. 8 shows the variation of the reduced Nusselt number with the nanoparticles volume fraction _{2}-nanoparticles and Al_{2}O_{3}-nanoparticles. Also, the Fig. 8 and Table 3 show that the values of

## 4. Conclusions

The problem of the steady free convection boundary layer flow past a vertical cone embedded in porous medium filled with a non-Newtonian nanofluid has been studied and the special case when the base fluid is water has been considered. The effects of the solid volume fraction _{2}O_{3}) and Titanium oxide (TiO_{2}). It has been shown, as expected, that increasing of the values of the nanoparticles volume fraction lead to an increase of the velocity and the temperature profiles and to an decrease of the Nusselt number for the values of the parameter